n-fold Hyperbolic Paraboloids and Related Structures

ABSTRACT

Utilizing the three different hyperbolic paraboloids which divide each of the defining tetrahedra (DT) in half, new geometric objects have been created which have unique stacking and interlocking characteristics and are inherently strong and rigid due to their triangular and hyperbolic paraboloid faces. These geometric objects can be utilized to build aesthetic and utilitarian components such as beams, trusses, packaging foams, toys, repeating cellular structures and others. The n-fold hyperbolic paraboloids are new geometric objects. The “n-” in the title stands for any integer greater than or equal to three. The fourfold hyperbolic paraboloid (FIG.  2 C) has the special attribute of being space filling. Like the cube the fourfold hyperbolic paraboloids can be continuously stacked so that there is no unenclosed volume between them. All of the n-fold hyperbolic paraboloids have unique stacking and interlocking attributes. The interlocking and stacking characteristics of these objects result from the saddle shaped compound curvature of the DT hyperbolic paraboloids. Three new three faced geometric objects or trihedrons (FIGS.  3 A  3 B, and  3 G) have been created from each DT, each has two DT isosceles triangular faces with the third face being one of the three hyperbolic paraboloids which divide the DT in half. These trihedrons have stacking, interlocking, and strength and rigidity characteristics similar to the n-fold hyperbolic paraboloids. Thickening the surfaces of the three hyperbolic paraboloids which divide the DT in half (FIGS.  4 B,  4 E, and  4 G) results in new thin geometric objects specifically defined by the DT. These new thin geometric objects can be used to build repeating cellular structures which effectively harness material properties to result in inherent strength and rigidity due to the hyperbolic paraboloid shape of the cell walls. They can also be applied to appurtenances of other geometric objects such as square bars, spheres, etc such that these other geometric objects can be joined in an interlocking fashion

CROSS-REFERENCE TO RELATED APPLICATIONS

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FEDERALLY SPONSORED RESEARCH

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SEQUENCE LISTING OR PROGRAM

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BACKGROUND

1. Field

This application relates to static geometric space defining structures, specifically to such structures which utilize hyperbolic paraboloids. In particular, this application relates to families of unique geometric objects which utilize the three different hyperbolic paraboloids that divide defining tetrahedra in half.

2. Prior Art

Structures such as roofs and tents which utilize hyperbolic paraboloids are in common use. Additionally the tetrahedron has been commonly used in trusses. The regular tetrahedron, which is one of the five platonic solids, is composed of four faces that are equilateral triangles, six edges between these faces, and four vertices where the corners of three triangular faces meet. This regular tetrahedron is not included in this application. The defining tetrahedra (DT) of this application all have four identical isosceles triangular faces.

Tetrahedrons have been used in trusses where a frame composed of structural members along each edge of the tetrahedron result in four vertices and multiple tetrahedrons are attached at their vertices resulting in a strong truss. What has not been recognized is the utility of using the hyperbolic paraboloids which divide the DTs of this application in half and whose curvature and edge dimensions are defined by the DT. Unexpectedly, these DT hyperbolic paraboloids can be utilized to make new, unique geometric objects which can be fit together in a joint that has superior interlocking attributes due to the saddle shaped compound curvature of the DT hyperbolic paraboloids. Thin geometric objects utilizing the hyperbolic paraboloids of the DT can be used to create cellular structures. All of the structures of this application effectively harness the superior rigidity resulting from the smooth saddle shaped compound curvature of the DT hyperbolic paraboloids. This curvature results in improved rigidity similar to the improved rigidity that results from a rolled up sheet of paper which is more rigid than a flat sheet. These hyperbolic paraboloids can be used to shape appurtenances of other geometric objects such as square bars, spheres, etc so that these other geometric objects can be joined in an interlocking fashion. The use of the DT hyperbolic paraboloids of this application results in a surprising variety of composite structures. There are three different hyperbolic paraboloids which are each defined by the DT, each divides the DT exactly in half and each can be used by themselves or as one or more faces of other geometric objects.

SUMMARY

A set of defining tetrahedrons (DT) can be established with each, in turn, defining three hyperbolic paraboloids used in constructing the new geometric objects of this application. Utilizing the three different hyperbolic paraboloids which divide each of the DTs in half, new geometric objects have been created which have unique stacking and interlocking characteristics and are inherently strong and rigid due to their triangular and hyperbolic paraboloid faces. The stacking and interlocking characteristics of these objects result from the saddle shaped compound curvature of the DT hyperbolic paraboloids. The “n-” in the title of this application is a variable which stands for any integer greater than or equal to three. The fourfold hyperbolic paraboloid (4hypar) of FIG. 2C has the special attribute of being space filling. Like the cube the new 4hypars can be continuously stacked so that there is no unenclosed volume between them. Thickening the surfaces of the three hyperbolic paraboloids which divide the DT in half (FIGS. 4B, 4E, and 4G) results in new thin geometric objects. These new thin geometric objects can be used to build repeating cellular structures which effectively harness material properties to result in inherent strength and rigidity due to the hyperbolic paraboloid shape of the cell walls. They can also be applied to appurtenances of other geometric objects such as square bars, spheres, etc such that these other geometric objects can be joined in an interlocking fashion

DRAWINGS—FIGURES

In the drawings, closely related figures have the same number but different alphabetic suffixes. Also when referring to a hidden edge or surface dashed leaders to the reference numerals are used.

FIG. 1A depicts a defining tetrahedron (DT) composed of four isosceles triangular faces which is prior art.

FIG. 2A illustrates the threefold hyperbolic paraboloid (3hypar).

FIG. 2B shows two 3hypars joined together and illustrates the alternating orientation of the axial vertices between adjacent 3hypars and that an axial vertex of one mates with a circumferential vertex of the other.

FIG. 2C illustrates the fourfold hyperbolic paraboloid (4hypar).

FIG. 2D shows four 4hypars joined together and illustrates the alternating orientation of the axial vertices between adjacent 4hypars and that an axial vertex of one mates with a circumferential vertex of an adjacent one.

FIG. 2E shows multiple 4hypars connected together in an arrangement that extends into all three spatial dimensions.

FIG. 2F illustrates the sixfold hyperbolic paraboloid (6hypar).

FIG. 2G shows three 6hypars joined together and illustrates the alternating orientation of the axial vertices between adjacent 4hypars and that an axial vertex of one mates with a circumferential vertex of an adjacent one.

FIG. 2H illustrates the eightfold hyperbolic paraboloid (8hypar).

FIG. 2I shows four 8hypars joined together and illustrates the alternating orientation of the axial vertices between adjacent 8hypars and that an axial vertex of one mates with a circumferential vertex of an adjacent one.

FIG. 3A illustrates a trihedron composed of two DT isosceles triangles and the DT hyperbolic paraboloid defined by the four L1 edges of the DT, termed the L1 hyperbolic paraboloid.

FIG. 3B illustrates the trihedron composed of two DT isosceles triangles and the right handed DT hyperbolic paraboloid defined by the two L2 DT edges and the two L1 DT edges which connect the two L2 edges in a right handed sense, termed the L2 right L1 hyperbolic paraboloid.

FIG. 3C illustrates the trihedron composed of two DT isosceles triangles and the left handed DT hyperbolic paraboloid defined by the two L2 DT edges and the two L1 DT edges which connect the two L2 edges in a left handed sense, termed the L2 left L1 hyperbolic paraboloid.

FIG. 3D illustrates how the three trihedrons of FIGS. 3A, 3B and 3C can be joined at their hyperbolic paraboloid faces, the result in each case being the DT.

FIG. 3E illustrates how the trihedron of FIG. 3A can be joined with the 4hypar of FIG. 2C.

FIG. 3F shows the result of joining the trihedron of FIG. 3A with the 4hypar of FIG. 2C, the result being a figure with two DT isosceles triangular faces and three L1 hyperbolic paraboloid faces.

FIG. 3G shows the result of joining four trihedrons of FIG. 3A with the 4hypar of FIG. 2C. The result is a dipyramid composed of eight DT isosceles triangular faces. This structure is composed of four DTs each having one L2 edge on the axial axis between the two axial vertices and the other L2 edge comprising one of four chords of the circle containing the four circumferential vertices. This dipyramid fills space.

FIG. 4A shows how the L1 hyperbolic paraboloid fits into the DT and is therefore defined by the DT.

FIG. 4B illustrates the thickened L1 hyperbolic paraboloid thickened surface.

FIG. 4C illustrates several L1 hyperbolic paraboloid thickened surfaces joined to make a cellular composite structure. One of the thickened L1 hyperbolic paraboloids has been highlighted in bold and is oriented similarly to those in FIGS. 4A and 4B.

FIG. 4D shows how the L2 right L1 hyperbolic paraboloid fits into the DT and is therefore defined by the DT.

FIG. 4E illustrates the thickened L2 right L1 hyperbolic paraboloid surface.

FIG. 4F shows how the L2 left L1 hyperbolic paraboloid fits into the DT and is therefore defined by the DT.

FIG. 4G illustrates the thickened L2 left L1 hyperbolic paraboloid surface.

DRAWINGS—REFERENCE NUMERALS

-   -   10 L1 DT edge     -   14 L2 DT edge     -   18 DT isosceles triangle consisting of one L2 edge and two L1         edges of equal length. The ratios of the length of an L1 edge to         the length of an L2 edge are a set of discrete values as         discussed in the text.     -   20 threefold hyperbolic paraboloid (3hypar)     -   21 fourfold hyperbolic paraboloid (4hypar)     -   23 sixfold hyperbolic paraboloid 6hypar)     -   25 eightfold hyperbolic paraboloid (8hypar)     -   L1 hyperbolic paraboloid     -   30 axial vertex     -   34 circumferential vertex     -   38 trihedron using the L1 hyperbolic paraboloid of the DT     -   40 trihedron using the L2 right L1 hyperbolic paraboloid of the         DT     -   42 trihedron using the L2 left L1 hyperbolic paraboloid of the         DT     -   44 Space filling dipyramid composed of 8 DT isosceles triangle         faces.     -   46 L2 right L1 hyperbolic paraboloid     -   50 L2 left L1 hyperbolic paraboloid

DETAILED DESCRIPTION First Embodiment E.g. FIGS. 2A, 2C, 2F, 2H

A set of defining tetrahedra (DT) can be established with each, in turn each defining three hyperbolic paraboloids used in constructing the new geometric objects of this application. A DT consist of four edges of one length (L1) and two edges of a second length (L2), with these edges then forming four identical isosceles triangular faces and four vertices where three of the isosceles triangular faces meet. The two L2 edges are opposite i.e. not adjacent to each other in the DT. The isosceles triangles are composed of one L2 and two L1 edges. Specific ratios of the length of an L1 edge to the length of an L2 edge are what set the proportions of the DTs and in turn their three hyperbolic paraboloids. The set of specific ratios are determined by the variable “n” as discussed below. An individual DT can be of any size, the only requirements being the ratio of an L1 edge to an L2 edge must be constant and the L2 edges must not be adjacent.

The “n” in the title of this application is a variable for the set of integers equal to or greater than three. Thus, this application includes the threefold hyperbolic paraboloid, the fourfold hyperbolic paraboloid, the fivefold hyperbolic paraboloid, the sixfold hyperbolic paraboloid, etc and related structures. For each “n” there is a different DT and therefore a different group of three hyperbolic paraboloids which divide the DT in half and are used to form the geometric objects of this application.

To further elaborate and quantify this, the specific DT L1/L2 ratios can be calculated as a function of “n”. These ratios then set the lengths and arrangements of the four equal length L1 edges and the two equal length L2 edges of the DT which in turn define the hyperbolic paraboloids that are used to create the geometric objects of this application. From the geometry of the DT and considering that one L2 edge will be the axial axis of the nhypar and the other L2 edge will be one of n chords of a circle setting the spacing of the circumferential vertices of the nhypar (e.g. FIG. 3G illustrates four such DTs placed adjacently such that one L2 edge of each DT is the axial axis between the two axial vertices and the other four L2 segments constitute chords of the circle containing the four circumferential vertices). The equation that defines L1/L2 as a function of n results from a geometrical analysis of the DT and is;

$\frac{L\; 1}{L\; 2} = \frac{\left\lbrack {1 + \left( \frac{1}{\sin \frac{360{^\circ}}{2n}} \right)^{2}} \right\rbrack^{\frac{1}{2}}}{2}$

The following table tabulates these L1/L2 values for the first several values of n.

N L1/L2 3 0.764 4 0.866 5 0.987 6 1.118 7 1.256 8 1.399 9 1.545 10 1.694 11 1.844 12 1.996 . . . . . .

Thus the specific DTs constructed from the above L1s and L2s define the hyperbolic paraboloids of this application.

FIG. 2A illustrates an example of the first embodiment, the threefold hyperbolic paraboloid (3hypar). The surface of the 3hypar consists of three L1 hyperbolic paraboloids connected at their edges thereby forming a closed volume. The L1 hyperbolic paraboloids are those whose curvature and edge lengths are defined by the arrangement of the four L1 DT edges. The 3hypar has inherent strength and rigidity characteristics due to the hyperbolic paraboloid shape of each of its three connected faces. In addition to these three faces the 3hypar has six edges (each the length of the L1 DT edge), two axial vertices and three circumferential vertices. The circumferential vertices lie in a plane midway between the axial vertices and the plane is perpendicular to the axis drawn between the axial vertices. With this configuration the distance between the axial vertices is exactly the same as the distance between adjacent circumferential vertices, a property which allows stacking of the 3hypars.

FIG. 2C illustrates another example of the first embodiment, the fourfold hyperbolic paraboloid (4hypar). The surface of the 4hypar consists of four L1 hyperbolic paraboloids connected at their edges thereby forming a closed volume. The 4hypar has inherent strength and rigidity characteristics due to the hyperbolic paraboloid shape of each of its four connected faces. The curvature of each of these hyperbolic paraboloids is defined by the arrangement of the four L1 DT edges. In addition to these four faces the 4hypar has eight edges (each the length of the L1 DT edge), two axial vertices and four circumferential vertices. The circumferential vertices lie in a plane midway between the axial vertices and the plane is perpendicular to the axis drawn between the axial vertices. With this configuration the distance between the axial vertices is exactly the same as the distance between adjacent circumferential vertices, a property which allows stacking of the 4hypars. Like the cube, 4hypars can be continuously stacked to fill space with no unenclosed space between them.

FIG. 2F illustrates another example of the first embodiment, the sixfold hyperbolic paraboloid (6hypar). The surface of the 6hypar consists of six L1 hyperbolic paraboloids connected at their edges thereby forming a closed volume. The 6hypar has inherent strength and rigidity characteristics due to the hyperbolic paraboloid shape of each of its six connected faces. The curvature of each of these hyperbolic paraboloids is defined by the arrangement of the four L1 DT edges. In addition to these six faces the 6hypar has twelve edges (each the length of the L1 DT edge), two axial vertices and six circumferential vertices. The circumferential vertices lie in a plane midway between the axial vertices and the plane is perpendicular to the axis drawn between the axial vertices. With this configuration the distance between the axial vertices is exactly the same as the distance between adjacent circumferential vertices, a property which allows stacking of the 6hypars.

FIG. 2H illustrates another example of the first embodiment, the eightfold hyperbolic paraboloid (8hypar). The surface of the 8hypar consists of eight L1 hyperbolic paraboloids connected at their edges thereby forming a closed volume. The 8hypar has inherent strength and rigidity characteristics due to the hyperbolic paraboloid shape of each of its eight connected faces. The curvature of each of these hyperbolic paraboloids is defined by the arrangement of the four L1 DT edges. In addition to these eight faces the 8hypar has sixteen edges (each the length of the L1 DT edge), two axial vertices and eight circumferential vertices. The circumferential vertices lie in a plane midway between the axial vertices and the plane is perpendicular to the axis drawn between the axial vertices. With this configuration the distance between the axial vertices is exactly the same as the distance between adjacent circumferential vertices, a property which allows stacking of the 8hypars.

FIGS. 2A, 2C 2F and 2H are only four of the many n-fold hyperbolic paraboloids (nhypars) that can be created. All nhypars use the L1 hyperbolic paraboloid for their faces, all form closed volumes, all have one L2 edge as the axial axis and the other L2 edge as one of n chords of the circle containing the circumferential vertices, and all of these geometric objects have 2n edges each of length L1.

Operation, First Embodiment E.g. FIGS. 2B, 2D, 2E, 2G, 2I

In FIG. 2B two 3hypars are stacked by placing the axial vertex of the first 3hypar between two adjacent circumferential vertices of the second 3hypar. This also results in the axial vertex of the second 3hypar being between two adjacent circumferential vertices of the first 3hypar. When joined in this manner the two axial axes and the four short SFT edges exactly coincide with the edges of the DT. More 3hypars could be added to the stack as desired.

In FIG. 2D four 4hypars are stacked by placing the axial vertex of the first 4hypar between two adjacent circumferential vertices of the second 4hypar and this is repeated two more times. This also results in the axial vertex of the next 4hypar being between two adjacent circumferential vertices of the earlier 4hypar. When joined in this manner the two axial axes and the four L1 DT edges at each joint exactly coincide with the edges of the DT. FIG. 2E shows twelve 4hypars stacked in a manner such that some extend into all three dimensions, a process that could be continued to completely fill space if desired.

In FIG. 2G three 6hypars are stacked by placing the axial vertex of the first 6hypar between two adjacent circumferential vertices of the second 6hypar and this is repeated one more time. This also results in the axial vertex of the next 6hypar being between two adjacent circumferential vertices of the earlier 6hypar. When joined in this manner the two axial axes and the four L1 DT edges at each joint exactly coincide with the edges of the DT. More 6hypars could be added to the stack as desired.

In FIG. 2I four 8hypars are stacked by placing the axial vertex of the first 8hypar between two adjacent circumferential vertices of the second 8hypar and this is repeated two more times.

This also results in the axial vertex of the next 8hypar being between two adjacent circumferential vertices of the earlier 8hypar. When joined in this manner the two axial axes and the four L1 DT edges at each joint exactly coincide with the edges of the DT. More 8hypars could be added to the stack as desired.

In Addition to the property which allows stacking of nhypars of equal dimensions, when a hyperbolic paraboloid surface from one nhypar is joined with the hyperbolic paraboloid surface of a second similar nhypar in a stack it interlocks. The two are then constrained against rotation about a common axis between them and the two are also constrained from sliding in either of the two dimensions perpendicular to the axis between them. This interlocking occurs for each additional nhypar added to the stack and additionally may completely restrain an earlier nhypar depending on where the next nhypar is added. Thus a stack of nhypars of any size (in increments of one nhypar) and without any means of attachment between the nhypars can be created having stability due to the interlocking characteristics of the nhypars. Further stability and strength of the composite stack can be achieved by providing a means to affix the nhypars to each other.

Affixing the nhypars results in the creation of strong structures. For example a long string of affixed nhypars could be used as a thin beam or a regular three dimensional lattice useful as a truss. An affixed planar stacking of selected nhypars could be assembled resulting in something that could be used as a wall, a floor, a ceiling, etc. Trusses of various configurations can be created by an appropriate stacking of affixed nhypars. A wide variety of similar useful stackings of affixed nhypars are possible. These nhypars could be useful by themselves for example as foam packaging material etc.

Second Embodiment E.g. FIGS. 3A, 3B, 3C

The second embodiment is the three families of trihedrons illustrated in FIGS. 3A, 3B, and 3C. Each of the trihedrons is composed of two DT isosceles triangular faces and a third face which is one of the three different hyperbolic paraboloid surfaces that divide the DT in half. Each of these trihedrons has the interlocking attributes achieved by stacking one trihedron to a copy of itself at their hyperbolic paraboloid faces. Each of these trihedrons has inherent strength and rigidity characteristics imbued by their triangular and hyperbolic paraboloid faces.

The family of trihedrons illustrated in FIG. 3A is composed of two planar SFT isosceles triangles with one L2 edge, two L1 edges and a third face being the L1 hyperbolic paraboloid. The edges and hyperbolic paraboloid are defined by the DT with the hyperbolic paraboloid being the one defined by the curvature and edges created by the four L1 DT edges.

The family of trihedrons illustrated in FIG. 3B is composed of two planar DT isosceles triangles and a third face being the L2 right L1 hyperbolic paraboloid. The edges and hyperbolic paraboloid are defined by the DT with the hyperbolic paraboloid being the one defined by the curvature and edges created by the two L2 DT edges and the two L1 DT edges that connect to the two L2 edges in a right handed sense, analogous to a right handed helix. This right handed sense attribute is elaborated further in the discussion of FIG. 4E below.

The family of trihedrons illustrated in FIG. 3C is composed of two planar DT isosceles triangles and a third face being the L2 left L1 hyperbolic paraboloid. The edges and hyperbolic paraboloid are defined by the DT with the hyperbolic paraboloid being the one defined by the curvature and edges created by the two L2 edges and the two L1 edges that connect to the two L2 edges in a left handed sense, analogous to a left handed helix. This left handed sense attribute is elaborated further in the discussion of FIG. 4G below.

Operation, Second Embodiment E.g. FIGS. 3D, 3E, 3F, 3G

Each of the three DT trihedrons can be combined with a similar trihedron as illustrated in FIG. 3D. In each case the result of the combination of two similar trihedrons is the DT. This result is to be expected when it is recalled that each of the three different hyperbolic paraboloids divides the DT exactly in half.

The trihedron of FIG. 3A has the special attribute that when n of them are affixed to each other at their triangular faces the nhypar is created. FIGS. 3E, 3F and 3G illustrate a sequence of attaching the trihedrons of FIG. 3A to the 4hypar of FIG. 2C by their hyperbolic paraboloid surfaces. FIG. 3E shows how one FIG. 3A trihedron can be attached to the 4hypar of FIG. 2C. FIG. 3F shows the resulting geometric object which has two SFT isosceles triangular faces and three L1 hyperbolic paraboloid faces. FIG. 3G illustrates the attachment of an additional three trihedrons of FIG. 3A to the 4hypar of FIG. 2C which results in a geometric object that is a square dipyramid with eight DT isosceles triangular faces. This square dipyramid fills space i.e. they can be stacked with no unenclosed volume between them.

Third Embodiment E.g. FIGS. 4B, 4E, 4G

The third embodiment is three thickened hyperbolic surfaces of each DT, the three hyperbolic paraboloid surfaces being the three which divide the DT in half. Because a surface is infinitesimally thin, thickening is required to provide a real three dimensional physical object that has structural integrity. These thickened surface geometric objects can be connected to each other at their corners and/or their edges to create a cellular structure which can be repeated indefinitely to create a composite cellular structure. The cell wall thickness would be just the thickness by which the hyperbolic paraboloid surface itself has been thickened. The compound curvature of the hyperbolic paraboloids results in a component that is more rigid than a flat planar surface of the same thickness analogous to the way a rolled up sheet of paper is more rigid than a flat sheet of paper.

FIG. 4B is the thickened L1 hyperbolic paraboloid surface whose hyperbolic paraboloid curvature and edge dimensions are defined by the four L1 edges of the DT.

FIG. 4E is the thickened L2 right L1 hyperbolic paraboloid surface whose hyperbolic paraboloid curvature and edge dimensions are defined by the two L2 edges of the DT and the two L1 DT edges that connect these two L2 edges in a right handed sense, analogous to a right handed helix. The handedness of the DT hyperbolic paraboloids which utilizes two L2 edges is determined by the following procedure. Place the thickened hyperbolic paraboloid on a flat surface in front of you with one L2 edge oriented from left to right. In this orientation one end of the second L2 edge will be on the flat surface toward the back and its other end will be elevated toward the front over the first L2 edge. If the L1 DT edge in the foreground which connects to the two L2 edges leans to the right as in FIG. 4E, it is a right handed hyperbolic paraboloid.

FIG. 4G is the thickened L2 left L1 hyperbolic paraboloid surface whose hyperbolic paraboloid curvature and edge dimensions are defined by the two L2 edges of the DT and the two L1 edges that connect these two L2 edges in a left handed sense, analogous to a left handed helix. The handedness of the DT hyperbolic paraboloids which utilize two L2 edges is determined by the following procedure. Place the thickened hyperbolic paraboloid on a flat surface in front of you with one L2 edge oriented from left to right. In this orientation one end of the second L2 edge will be on the flat surface toward the back and its other end will be elevated toward the front over the first L2 edge. If the L1 DT edge in the foreground which connects to the two L2 edges leans to the left as in FIG. 4G, it is a left handed hyperbolic paraboloid

Operation, Third Embodiment E.g. FIGS. 4A, 4C, 4D, 4F

FIGS. 4A, 4D, and 4F illustrate how the three hyperbolic paraboloids which divide the DT in half fit in the DT with FIGS. 4B, 4E, and 4G being those same corresponding hyperbolic paraboloids thickened and outside the DT frame. FIG. 4A illustrates how the L1 hyperbolic paraboloid fits into the DT frame and is thus defined by the DT; note that the two L2 DT edges are not a part of this L1 hyperbolic paraboloid. FIG. 4B illustrates the thickened L1 hyperbolic paraboloid surface. FIG. 4C illustrates how several thickened hyperbolic paraboloids of FIG. 4B can be attached to form a composite cellular structure whose cell walls are thickened L1 hyperbolic paraboloids. Note that one of the thickened L1 hyperbolic paraboloids of FIG. 4C has been highlighted in bold and is shown in the same orientation as those in FIGS. 4A and 4B.

FIG. 4D illustrates how the L2 right L1 hyperbolic paraboloid fits into the DT frame and is thus defined by the DT; note that the two L1 DT edges that connect to the L2 edges in a left handed sense are not a part of the L2 right L1 hyperbolic paraboloid. FIG. 4E illustrates the thickened L2 right L1 hyperbolic paraboloid surface.

FIG. 4F illustrates how the L2 left L1 hyperbolic paraboloid fits into the DT frame and is thus defined by the DT; note that the two L1 DT edges that connect to the L2 edges in a right handed sense are not a part of the L2 left L1 hyperbolic paraboloid. FIG. 4G illustrates the thickened L2 left L1 hyperbolic paraboloid surface.

Each of the three types of thickened hyperbolic paraboloid surfaces could be attached to themselves or each other at their corners since all corners coincide with vertices of the DT. Also each thickened hyperbolic paraboloid surface would have two edges which coincide with two edges of each of the other two thickened hyperbolic paraboloid surfaces.

The multi-celled composite structures that can be created by the connection of the three thickened hyperbolic paraboloid surfaces are unique and have aesthetic and utilitarian uses. The walls of all cells are hyperbolic paraboloids. The multi-celled composite structures effectively harness the superior rigidity and strength resulting from the smooth compound curvature of the DT hyperbolic paraboloids.

The hyperbolic paraboloids of FIGS. 4B, 4E, and 4G could be placed onto other geometric bodies thereby allowing them to be joined. As one specific example the hyperbolic paraboloid of FIG. 4B when viewed along an axis through its null point in the center has a square silhouette and thus this hyperbolic paraboloid could be placed onto the end of a long bar with a square cross section and two such bars could then be connected in an interlocking manner at their hyperbolic paraboloid ends. Other examples would include putting these hyperbolic paraboloids onto bosses extending from the surfaces of spheres at six points such that the three axes drawn between pairs of points are mutually orthogonal. This would then allow connection of these spheres in an interlocking fashion in a three dimensional square array.

ADVANTAGES

From the description above a number of advantages of these presently preferred embodiments become evident:

-   -   Because the proportions of the geometric objects and the         curvature of their hyperbolic paraboloids are those defined by         the DT, these objects can be stacked.     -   The fourfold hyperbolic paraboloid fills space, i.e. they can be         stacked continuously with no unenclosed volume between them.     -   The triangular and hyperbolic faces imbue these geometric         objects and the composite structures created from them with         inherent structural rigidity and strength.     -   The nfold hyperbolic paraboloid families and their three         trihedron families are new geometric shapes.     -   When the nfold hyperbolic paraboloid and trihedron geometric         objects are joined at their hyperbolic paraboloid faces they         become interlocked due to the saddle shaped compound curvature         of the hyperbolic paraboloids.     -   The families of thickened hyperbolic paraboloids are new         geometric objects specifically defined by the DT and they can be         used to create new cellular structures with hyperbolic         paraboloid shaped walls.     -   The thickened hyperbolic paraboloid surfaces have an inherent         rigidity due to the compound curvature of the hyperbolic         paraboloid similar to the increased rigidity obtained by rolling         up a sheet of paper.     -   The thickened L1 hyperbolic paraboloids can be continuously         connected to create a cellular structure that repeats throughout         space.     -   The hyperbolic paraboloids described herein could be affixed to         other geometric objects such as square bars or spheres allowing         these other geometric objects to be joined in an interlocking         fashion.

CONCLUSIONS, RAMIFICATIONS, AND SCOPE

Accordingly the new geometric objects of the various embodiments can be used to create new aesthetic and utilitarian structures. Furthermore these geometric objects have additional advantages:

-   -   They can be made to be any size by scaling the dimensions         proportionally with the DT which define their edge lengths and         hyperbolic paraboloid curvatures.     -   They can be made of metal, plastic, foam, ceramic, glass, wood,         masonry or other materials.     -   They can be made in various colors or color combinations.     -   They can be easily modified to provide a means of attachment to         each other or to other structural components. The potential         means of attachment include something as simple as putting a         hole through them for attachment hardware or the use of         adhesives.     -   They are modular allowing creation of composite structures.     -   These geometric objects could be used as toys or parts of toys.

Although the descriptions above contain many specifics, these should not be construed as limiting the scope of the embodiments but as merely providing illustrations of some of the presently preferred embodiments. For example, another embodiment of value is a three dimensional latticework of nhypars which results from the connection of multiple nhypars at their respective axial vertices and then also making additional connections at selected circumferential vertices.

Thus the scope of the embodiments should be determined by the appended claims and their legal equivalents, rather than by the examples given. 

1. A family of geometric objects (e.g. FIGS. 2A, 2C, 2F, and 2H) comprising sections of n hyperbolic paraboloid surfaces arranged symmetrically about an axial axis with curvature of said hyperbolic paraboloid surfaces defined by the arrangement of the four L1 edges of the defining tetrahedron wherein said n is any integer equal to or greater than three wherein said sections of n hyperbolic paraboloid surfaces can be extended to intersect at 2n edges each of length and position predetermined by said L1 edges of said defining tetrahedrons wherein said n sections of hyperbolic paraboloid surfaces can be extended to result in said hyperbolic surfaces intersecting at two axial and said n circumferential vertices whereby multiple units of said geometric objects being stackable in an interlocking fashion.
 2. The geometric object of claim 1 wherein the interior of said geometric objects is solid, is a void, or is composed of a frame or cellular structure.
 3. The geometric object of claim 1 wherein said sections of n hyperbolic paraboloid surfaces have been extended to intersect at said 2n edges of length and position predetermined by said L1 edges of said defining tetrahedrons wherein said sections of n hyperbolic paraboloid surfaces have been extended to intersect at two axial and n circumferential vertices.
 4. The geometric object of claim 3 wherein said 2n L1 edges and/or said axial vertices, and/or said n circumferential vertices, and/or said sections of n hyperbolic surfaces have been modified to provide means for attaching said geometric object to adjacent said geometric objects or other structural components.
 5. A geometric object (FIG. 3A) comprising two planar defining tetrahedron isosceles triangular faces and a third face consisting of a section of a hyperbolic paraboloid surface with the curvature of said section of a hyperbolic paraboloid surface being defined by the arrangement of the four L1 edges of said defining tetrahedron wherein said section of a hyperbolic paraboloid surface and said two planar defining tetrahedron isosceles triangular faces can be extended to said four L1 edges of length and position predetermined by said defining tetrahedron wherein said two planar isosceles triangular face extension results in a fifth edge which is an L2 edge of said defining tetrahedron and said fifth edge is not a part of said section of a hyperbolic paraboloid surface whereby multiple units of said geometric object are stackable in an interlocking fashion at the interface of said hyperbolic paraboloid surfaces and stackable at the interfaces of said two planar defining tetrahedron isosceles triangular faces.
 6. A geometric object (FIG. 3B) comprising two planar defining tetrahedron isosceles triangular faces and a third face consisting of a section of a right handed hyperbolic paraboloid surface with the curvature of said section of a right handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of said defining tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a right handed sense wherein said two planar defining tetrahedron isosceles triangular faces and said section of a right handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron wherein the extension of said two planar defining tetrahedron isosceles triangular faces results in a fifth edge wherein said fifth edge which is an L1 edge of length and position predetermined by said defining tetrahedron and said fifth edge is not a part of said section of a right handed hyperbolic paraboloid surface whereby multiple units of said geometric object are stackable in an interlocking fashion at the interface of said right handed hyperbolic paraboloid surfaces and stackable at the interfaces of said two planar defining tetrahedron isosceles triangular faces.
 7. A geometric object (FIG. 3C) comprising two planar defining tetrahedron isosceles triangular faces and a third face consisting of a section of a left handed hyperbolic paraboloid surface with the curvature of said section of a left handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of said defining tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a left handed sense wherein said two planar defining tetrahedron isosceles triangular faces and said section of a left handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron wherein the extension of said two planar defining tetrahedron isosceles triangular faces results in a fifth edge wherein said fifth edge which is an L1 edge of length and position predetermined by said defining tetrahedron and said fifth edge is not a part of said section of a left handed hyperbolic paraboloid surface whereby multiple units of said geometric object are stackable in an interlocking fashion at the interface of said left handed hyperbolic paraboloid surfaces and stackable at the interfaces of said two planar defining tetrahedron isosceles triangular faces.
 8. The geometric object of claims 5, 6, and 7 wherein the interior of said geometric object is solid or is composed of a frame or cellular structure.
 9. The geometric object of claims 5, 6, and 7 (FIGS. 3A, 3B, 3C) wherein said two planar defining tetrahedron isosceles triangular faces and said section of hyperbolic paraboloids have been extended to intersect at five edges of length and position predetermined by said defining tetrahedron.
 10. The geometric object of claims 5, 6, and 7 (FIGS. 3A, 3B, 3C) wherein said L1 edges and/or said L2 edges and/or said sections of hyperbolic paraboloids and/or said two planar defining tetrahedron isosceles triangular faces have been modified for design purposes e.g. to provide means for attaching said geometric objects to adjacent said geometric objects or other structural components.
 11. A geometric object (FIG. 4B) comprising a section of a thickened hyperbolic paraboloid surface with the curvature of said section of a thickened hyperbolic paraboloid surface predetermined by the arrangement of the four L1 edges of the defining tetrahedron wherein said section of a thickened hyperbolic paraboloid surface can be extended to said four L1 edges of length and position predetermined by said defining tetrahedron whereby multiple units of said sections of thickened hyperbolic paraboloid surfaces can be connected at their edges and/or corners.
 12. A geometric object (FIG. 4E) comprising a section of a thickened right handed hyperbolic paraboloid surface with the curvature of said section of a thickened right handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of the defining tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a right handed sense wherein said section of a thickened right handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron whereby multiple units of said sections of thickened right handed hyperbolic paraboloid surfaces can be connected at their edges and/or corners.
 13. A geometric object (FIG. 4G) comprising a section of a thickened left handed hyperbolic paraboloid surface with the curvature of said section of a thickened left handed hyperbolic paraboloid surface predetermined by the arrangement of the two L2 edges and two L1 edges of the space filling tetrahedron wherein said two L1 edges are those which connect to said two L2 edges in a left handed sense wherein said section of a thickened left handed hyperbolic paraboloid surface can be extended to said two L2 edges and said two L1 edges of length and position predetermined by said defining tetrahedron whereby multiple units of said sections of thickened left handed hyperbolic paraboloid surfaces can be connected at their edges and/or corners.
 14. The geometric objects of claims 11, 12, and 13 (FIGS. 4B, 4E, 4G) wherein said sections of thickened hyperbolic surfaces have been extended to said L2 and said L1 edges of length and position predetermined by said defining tetrahedron.
 15. The geometric objects of claims 11, 12, and 13 (FIGS. 4B, 4E, 4G) wherein said L1 edges and/or said L2 edges and/or corners and/or said sections of thickened hyperbolic paraboloid surfaces have been modified for design purposes e.g. to provide means for attaching said geometric objects to adjacent said geometric objects or other structural components. 